In nonparametric smoothing, the roughness penalty approach is widely used when smoothing noisy data [1]}, [2]}, [3]}. Including a roughness penalty and choosing a proper tuning parameter can avoid overfitting problems and balances the bias and variance of the estimator of \(m\) . We formulate the roughness penalty approach as the following penalized least squares problem:
\(\min _{m} \sum _{i=1}^n\left\lbrace Y_i-m(v_{i})\right\rbrace ^{2}+\lambda \mathcal {E}(m),\) 
